Computation of limit periodic continued fractions. A survey

Over the last 20 years a large number of algorithms has been published to improve the speed and domain of convergence of continued fractions. In this survey we show that these algorithms are strongly related. Actually, they essentially boil down to two main principles.We also prove some results on asymptotic expansions of tail values of limit periodic continued fractions.Dedicated to Luigi Gatteschi on his seventieth birthdayThis research was partially supported by The Norwegian Research Council and by the HMC project ROLLS, under contract CHRX-CT93-0416.     

On the group extension of the transformation associated to non-archimedean continued fractions

Summary Let F_q be a finite field with q elements. We consider formal Laurent series of F_q -coefficients with their continued fraction expansions by F_q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.     

柴油烃族组成分布的GC-MS测定

应用GC－MS测定柴油烃族组成按沸点的分布，通过柱色 谱分离后的柴油饱和烃和芳烃组分分别进入气相色谱－质谱联用仪分析，采集其每一扫描的质谱图后，按ASTM－D2425方法计算其每一扫描的烃族组成，因为每一扫描与保留时间对应，所以可将两部分烃族组成加和后应用ASTM－D2887方法计算柴油烃族组成按沸点的分布规律；实验结果表明，该法与ASTM－D2887和ASTM－D2425的实验结果吻合，并能给出有关柴油烃族组成的详尽分布规律。     

Gel permeation chromatographic characterization of sodium hyaluronate and its fractions prepared by ultrasonic degradation

Summary High-molecular-weight sodium hyaluronate isolated from rooster combs was degraded by ultrasonication. The molecular weight of hyaluronate and its polydispersity was determined by gel-permeation chromatography. During 75-min treatment the molecular weight value decreased from 1.39×10⁶ Da to 2.25×10⁵ Da while the polydispersity of the molecular weight increased from 1.29 to 2.36. The reciprocal value of the square of the hyaluronate's molecular weight was linearly proportional to the time of ultrasonication.     

Size exclusion chromatographic characterization of sodium hyaluronate fractions prepared by high energetic sonication

Summary A high molecular weight sodium hyaluronate isolated from rooster combs was degraded by ultrasonication. High-performance size exclusion chromatography allowed rapid and accurate determination of molecular weight parameters (M_z, M_w, M_n) and distributions. The time dependence of hyaluronan ultrasonication to the molecular characteristics of the polymer was investigated. A non-random nature of the degradation process was demonstrated and the reciprocal M_n value was found to be linearly proportional to the time of ultrasonication.     

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Characterizations of generalized Hermite and sieved ultraspherical polynomials

A new characterization of the generalized Hermite polyno-
mials and of the orthogonal polynomials with respect to the measure
is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.

     

Molecular properties of hemicelluloses located in the surface and inner layers of hardwood and softwood pulps

The molecular properties of hemicelluloses located in the surface and inner layers of fibers present in hardwood and softwood pulps, together with the effects of different bleaching processes on these properties, have been investigated in this study. In order to separate the hemicelluloses located in these two layers, fibers were subjected to mechanical peeling and then separated by filtration into surface (filtrate) and inner layer materials. The materials thus obtained were characterized with respect to their polysaccharide compositions and uronic acid contents. The molar mass parameters of the hemicelluloses (extracted by alkali) were determined by employing size-exclusion chromatography in combination with off-line MALDI mass spectrometry. For all of the pulps examined, the relative content of xylan was found to be greater in the surface layer of the fiber than in the corresponding inner layer. The xylan polymers of the surface layer exhibited higher molar masses and lower frequencies of uronic acid side groups than did the xylans in the inner fiber layer. In connection with ozone treatment, hexenuronic acid residues in the surface layer xylan were removed to a greater extent than in the case of the inner layers, indicating a gradient for the reaction with ozone across the fiber wall. The xylan polymer remaining on the surface of the softwood pulps after completion of the chlorine dioxide bleaching process was predominantly uncharged.     

Syntheses and structures of six compounds in the A2LiMS4 (A=K, Rb, Cs; M=V, Nb, Ta) family

Six new compounds in the A₂LiMS₄ (A=K, Rb, Cs; M=V, Nb, Ta) family, namely K₂LiVS₄, Rb₂LiVS₄, Cs₂LiVS₄, Rb₂LiNbS₄, Cs₂LiNbS₄, and Rb₂LiTaS₄, have been synthesized by the reactions of the elements in Li₂S/S/A₂S₃ (A=K, Rb, Cs) fluxes at 773 K. The A and M atoms play a role in the coordination environment of the Li atoms, leading to different crystal structures. Coordination numbers of Li atoms are five in K₂LiVS₄, four in A₂LiVS₄ (A=Rb, Cs) and Cs₂LiNbS₄, and both four and five in Rb₂LiMS₄ (M=Nb, Ta). The A₂LiVS₄ (A=Rb, Cs) structure comprises one-dimensional chains of tetrahedra. The Rb₂LiMS₄ (M=Nb, Ta) structure is composed of two-dimensional layers. The Cs₂LiNbS₄ structure contains one-dimensional chains that are related to the Rb₂LiMS₄ layers. The K₂LiVS₄ structure contains a different kind of layer.     

Semiempirical Molecular Orbital Studies of the Acylation Step in the Lipase‐Catalyzed Ester Hydrolysis

In this study, we present the results from the semiempirical molecular orbital calculations for the acylation step in the lipase‐catalyzed ester hydrolysis. The results reveal that the lowest energy path for the formation of the tetrahedral intermediate is for the serine residue of the catalytic triad to attack the substrate, followed by coupling heavy atom movement and proton transfer. The calculations of four active site models show that the cooperation of the aspartate group and the oxyanion hole is capable of lowering the activation energy by about 16 kcalmol^?1. Our results further suggest that the lipase‐catalyzed ester hydrolysis adopts the single proton transfer mechanism.